The converse of the Pythagorean theorem is also a true...

The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle. The points $(-3,4),(2,-1),(5,2),$ and $(0,7)$ are vertices of a quadrilateral. Show that the quadrilateral is a rectangle. (Hint: Show that the quadrilateral's opposite sides are the same length and that the two diagonals are the same length.)

The converse of the Pythagorean theorem is also a true statement: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Use the distance formula and the Pythagorean theorem to determine whether the set of points could be vertices of a right triangle.
The points $(-3,4),(2,-1),(5,2),$ and $(0,7)$ are vertices of a quadrilateral. Show that the quadrilateral is a rectangle. (Hint: Show that the quadrilateral's opposite sides are the same length and that the two diagonals are the same length.)

Key Concepts

Properties of Right Triangles

Right triangles are characterized by one 90-degree angle, which results in unique properties. In such triangles, the Pythagorean theorem provides a method to find missing side lengths, and the identification of a right angle can be based on the relationship between the squares of the side lengths. These properties are pivotal when verifying whether a triangle formed by given points is indeed right-angled.

Distance Formula

The distance formula is an essential tool in coordinate geometry used to calculate the distance between two points in the Cartesian plane. Derived from the Pythagorean theorem, it involves taking the square root of the sum of the squared differences of the coordinates. This formula is instrumental in determining side lengths and diagonal measures of geometric figures defined by their vertices.

Properties of Rectangles in Coordinate Geometry

Rectangles are quadrilaterals with four right angles. In coordinate geometry, a quadrilateral is verified as a rectangle if it has congruent and parallel opposite sides, as well as congruent diagonals. Checking for these properties—such as using the distance formula to measure which sides are equal and ensuring the diagonals have the same length—provides a systematic method to confirm the figure is a rectangle.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that establishes a relationship between the three sides of a right triangle. It states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. This theorem serves as the basis for many other geometric and algebraic concepts, especially in problems involving distances and angles.

Converse of the Pythagorean Theorem

The converse of the Pythagorean theorem asserts that if a triangle has side lengths that satisfy the equation a² + b² = c², then the triangle is a right triangle. This is particularly useful in determining whether a given triangle is right-angled when only the side lengths are known, without initially knowing the measure of any angles.

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